Octahedron in Tetrahedron

The applet below illustrates the fact that a tetrahedron is a union of four smaller tetrahedra and one octahedron. This fact (observed by Matt Henderson) leads to a solution to the problem of comparing the volumes of a regular tetrahedron and a square pyramid, with all 14 edges of the two shapes equal.

Unlike the dual embedding of an octahedron into a cube, now the vertices of the octahedron are located at the midpoint of the edges of the tetrahedron, and the edges of the former are the midlines of the latter.